DALLAS, Texas (CNN) -- The family of Timothy Cole, a Texas man who died in prison nearly a decade ago while serving a sentence for a rape he swore he did not commit, is hoping a court will issue the state's first posthumous exoneration.

Timothy Cole died in prison while serving a sentence for a rape DNA tests show he did not commit.

Cole was convicted and sentenced to 25 years in prison for the 1985 rape of 20-year-old Michele Mallin. He maintained his innocence, but it was not confirmed by DNA until years after his 1999 death, when another inmate confessed to the rape.

The court hearing on the exoneration began Thursday afternoon, according to the Austin American-Statesman.

Mallin, who has spoken publicly about the case, will join Cole's family in the Austin, Texas, courtroom. They want a judge to clear Cole's name, according to the Innocence Project of Texas, a nonprofit organization that seeks to help the wrongfully convicted.

Then a student at Texas Tech University in Lubbock, Mallin was walking to her car, intending to move it to another parking lot, when a man approached her asking about jumper cables, she told CNN. In a matter of seconds, he put her in a chokehold, a knife to her neck. He forced himself into her car and drove her to the outskirts of town, where he raped her.

The next day, police investigators showed Mallin pictures of possible suspects. She chose a picture of Cole and said he was her attacker. She later identified him in a physical lineup, according to the Innocence Project of Texas.

"I was positive," she told CNN. "I really thought it was him."

But there was one detail: Mallin told police her attacker was a smoker. "He was smoking the entire time."

And Cole, who suffered from severe asthma, "was never a smoker," Session said. "He took daily medications (for asthma) when he was younger."

"He was the sacrificial lamb. To them, my brother was the Tech rapist, there was no backtracking. It was the trial of the decade for Lubbock."

The "Tech rapist" attacked four women other than Mallin -- abducting them in parking lots near campus and driving them to a vacant location, where he would rape them and flee on foot, according to the Innocence Project of Texas. The rapist "terrorized" the Texas Tech campus in the mid-1980s, the organization said.

Cole, like Mallin, was a student at Texas Tech. He had finished two years of college previously and was returning to school after spending two years in the Army, his brother said.

But his dreams of getting married and having children never materialized. He was arrested and charged with Mallin's rape, declining a plea bargain offer that would have put him on probation. A jury convicted him and imposed a 25-year sentence.

That night, "he hugged my mother, and he said, 'Mother, why these people lie on me? Why they do this to me?' " Cole's brother Reggie Session recounted for the Lubbock Avalanche-Journal, which published a three-part series on the case in June.

"He said, 'They know I ain't done nothing to that girl. I don't even know that girl. Why they do this to me, mother?' ... He cried in my mother's arms on the floor."

Later, while in prison, Cole rejected an offer of parole that would have required him to admit his guilt. "His greatest wish was to be exonerated and completely vindicated," his mother, Ruby Session, told CNN affiliate News 8 Austin.

But the asthma that plagued Cole throughout his life brought about his death on December 2, 1999. The cause was determined to be heart complications due to his asthmatic condition. He was 39.

It was 2007 when a letter addressed to Cole arrived at his family's home, written by Johnson. Read the letter »

"You may recall my name from your 1986 rape trial in Lubbock," says the letter, dated May 11, 2007. "Your Lubbock attorney, Mike Brown, tried to show I committed the rape.

"I have been trying to locate you since 1995 to tell you I wish to confess I did in fact commit the rape Lubbock wrongly convicted you of. It is very possible that through a written confession from me and DNA testing, you can finally have your name cleared of the rape ... if this letter reaches you, please contact me by writing so that we can arrange to take the steps to get the process started. Whatever it takes, I will do it."

Johnson did not know Cole had died. In fact, according to the Avalanche-Journal, he had been writing to court officials for years to confess to the rape, but got nowhere.

Upon finding out that Cole was dead, Johnson wrote he "cried and felt double guilty, even though I know the system's at fault," according to the Avalanche-Journal.

"A day later, I am still bothered, terribly, by the death revelation. Because, not knowing Mr. Cole at all, I wonder if the wrongful incarceration contributed to his death."

Johnson has been in prison since 1985 on two convictions for aggravated sexual assault, according to the Texas Department of Corrections. He was given a life sentence for the rape of a 15-year-old girl, and a jury later tacked on a 99-year sentence for another rape, according to the Avalanche-Journal. He cannot be charged with the Mallin case, as the statute of limitations has expired.

The Innocence Project became involved after Cole's family received Johnson's letter. DNA tests confirmed that Johnson was Mallin's attacker. Now, Cole's family hopes the court hearing will be the final step in clearing his name.

Mallin is helping them. "I was very traumatized," she told CNN. "I was scared for my life. I tried my hardest to remember what he looked like.